The improvement in the positioning accuracy of machine tools necessitates reliable friction models for compensation. Friction and damping are primarily caused by mechanical contacts, and they have a wide influence on the dynamics of machine tools. Particularly in the linear motor driven axis, linear bearings induce majority of the friction; contact is observed between the ball and raceway in linear bearings. Based on the Hertzian contact theory and a tangential force model, a model is developed for the friction behavior during the contact between the ball and raceway. This model determines the stick and slip areas, and the relative velocity at the contact surface. Hence, the calculation of the friction force, its hysteresis characteristics, and the stick and slip portions becomes possible.

1. [1] S. Ito, “Analysis of rolling friction between steel ball and raceway,” Bearing Engineer, Vol.6, No.1, pp. 784-792, 1957 (in Japanese).
2. [2] T. Steinert, “Friction torque of ball bearings with retainer,” Ph.D. Thesis, Rheinisch-Westfälische Technische Hochschule Aachen (RWTH Aachen), 1996 (in German).
3. [3] R. Teutsch, “Contact models and strategies for simulation of rolling bearings and linear motion guides,” Ph.D. Thesis, University of Kaiserslautern, 2005 (in German).
4. [4] T. Fujita, A. Matsubara, and K. Yamazaki, “Experimental characterization of disturbance force in a linear drive system with high precision rolling guideways,” Int. J. Mach. Tools Manuf., Vol.51, No.2, pp. 104-111, 2011.
5. [5] T. Tanaka, J. Otsuka, and T. Oiwa, “Precision positioning control by modeling frictional behaviors of linear ball guideway,” Int. J. Automation Technol., Vol.3 No.3, pp. 334-342, 2009.
6. [6] F. Al-Bender, V. Lampaert, and J. Swevers, “The Generalized Maxwell-Slip Model: A Novel Model for Friction Simulation and Compensation,” IEEE Trans. on Automatic Control, Vol.50, No.11, pp. 1883-1887, 2005.
7. [7] F. Al-Bender and W. Symens, “Characterization of frictional hysteresis in ball-bearing guideways,” Wear, Vol.258, No.11-12, pp. 1630-1642, 2005.
8. [8] S. Futami, A. Furutani, and S. Yoshida, “Nanometer positioning and its micro-dynamics,” Nanotechnology, Vol.1, No.1, pp. 31-37, 1990.
9. [9] B. Armstrong-Hélouvry, P. Dupont, and C. C. de Wit, “A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction,” Automatica, Vol.30, No.7, pp. 1083-1138, 1994.
10. [10] A. Matsubara, A. Sayama, T. Sakai and M. Reuss, “Analysis of Measured Friction of Rolling Balls in Raceway Grooves,” Int. J. Automation Technol., Vol.8, No.6, pp. 811-819, 2014.
11. [11] J. J. Kalker, “Three-Dimensional Elastic Bodies in Rolling Contact – Solid Mechanics and its Applications,” Kluwer Academic publishers, pp. 28-34, 1990.
12. [12] K. L. Johnson, “Contact Mechanics,” Cambridge University Press, pp. 84-106/p. 427, 1985.
13. [13] V. L. Popov, “Contact Mechanics and Friction – Physical Principles and Applications,” Springer, Chapter 5, 2010.
14. [14] D.-F. Hu, Y.-H. Ji, and Y. Zhao, “Contact static stiffness research on machine tool considering the contact surface of rolling guideways,” Mechatronics and Manufacturing Technologies: Proc. of the Int Conf. on Mechatronics and Manufacturing Technologies (MMT2016), pp. 196-203, 2016.
15. [15] T. Stolarski, “Tribology in Machine Design,” Butterworth-Heinemann, pp. 249-250, 1999.
16. [16] N. Soda, Y. Kimura, and M. Sekizawa, “Wear occurring in rolling-sliding contact,” Proc. of The Japan Society of Machine Tool Engineers, Vol.37, p. 2204, 1970 (in Japanese).
17. [17] Y. Kimura, M. Sekizawa, and A. Nitanai, “Wear and fatigue in rolling contact,” Wear, Vol.253, No.1-2, pp. 9-16, 2002.
18. [18] T. Fujita, A. Matsubara, and S. Yamada, “Analysis of Friction in Linear Motion Rolling Bearing with Locomotive Multi-Bristle Model Influence of Slipping Velocity Distribution on Friction Characteristics,” Trans. of The Japan Society of Mechanical Engineers, Vol.77, Issue 778, pp. 2486-2495, 2011 (in Japanese).
19. [19] W. Beitz, K.-H. Küttner, and B. J. Davis (Eds.), “Dubbel – handbook of mechanical engineering,” Springer, p. E72, 1994 (in German).
20. [20] J. Awrejcewicz, “Classical Mechanics, Kinematics and Statics,” Springer, p. 79, 2012.